Advanced Linear Algebra Calculator

Master linear algebra with our powerful calculator. Compute matrices, vectors, eigenvalues, determinants, and more with step-by-step solutions.

Linear Algebra Calculator

Our comprehensive linear algebra calculator handles a wide range of operations on matrices and vectors. Select the operation you need below to get started.

Matrices
Vectors
Linear Independence
Vector Spaces
Eigenvalues & Eigenvectors

Matrix Operations

Perform various operations on matrices, including addition, subtraction, multiplication, determinant, inverse, and more.

Matrix A

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Examples

Matrix Properties
Determinant
Matrix Inverse
Matrix Multiplication
Row Reduced Echelon Form

Vector Operations

Perform various operations on vectors, including addition, subtraction, dot product, cross product, and more.

Vector A

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Vector Properties
Dot Product
Cross Product
Angle Between Vectors
Vector Projection

Linear Independence Calculator

Check if a set of vectors is linearly independent or linearly dependent.

Result

Step-by-Step Solution

Examples

Linearly Independent Vectors
Linearly Dependent Vectors
Complex Vectors
Symbolic Vectors

Vector Spaces Calculator

Compute properties of vector spaces, including basis, dimension, row space, column space, and null space.

Matrix

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Find Basis
Row Space
Null Space
Span of Vectors
Orthogonal Basis

Eigenvalues & Eigenvectors Calculator

Calculate eigenvalues, eigenvectors, characteristic polynomial, diagonalization, and more.

Matrix

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Eigenvalues & Eigenvectors
Characteristic Polynomial
Diagonalization
Jordan Normal Form
Spectral Decomposition

Linear Algebra Concepts and Formulas

Understanding the fundamental concepts and formulas of linear algebra is essential for solving problems. Here are the most important concepts with examples:

Matrix Operations

\text{Addition: } A + B = [a_{ij} + b_{ij}] \\ \text{Subtraction: } A - B = [a_{ij} - b_{ij}] \\ \text{Scalar Multiplication: } cA = [c \cdot a_{ij}] \\ \text{Matrix Multiplication: } (AB)_{ij} = \sum_{k=1}^{n} a_{ik} \cdot b_{kj}

These are the basic operations for manipulating matrices.

Example: If $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}$, then $A + B = \begin{bmatrix} 6 & 8 \\ 10 & 12 \end{bmatrix}$

Determinant

\det(A) = |A| = \sum_{j=1}^{n} a_{1j} \cdot C_{1j}

The determinant is a scalar value calculated from a square matrix, useful for finding inverses and solving systems of equations.

Example: $\det\begin{bmatrix} a & b \\ c & d \end{bmatrix} = ad - bc$

Matrix Inverse

A^{-1} = \frac{1}{\det(A)} \cdot \text{adj}(A)

The inverse of a matrix A is a matrix that, when multiplied by A, gives the identity matrix.

Example: If $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$, then $A^{-1} = \begin{bmatrix} -2 & 1 \\ 1.5 & -0.5 \end{bmatrix}$

Vector Operations

\text{Addition: } \vec{a} + \vec{b} = [a_i + b_i] \\ \text{Scalar Multiplication: } c\vec{a} = [c \cdot a_i] \\ \text{Dot Product: } \vec{a} \cdot \vec{b} = \sum_{i=1}^{n} a_i b_i \\ \text{Cross Product: } \vec{a} \times \vec{b} = \begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}

These operations allow manipulation and measurement of vectors.

Example: If $\vec{a} = [1, 2, 3]$ and $\vec{b} = [4, 5, 6]$, then $\vec{a} \cdot \vec{b} = 1 \cdot 4 + 2 \cdot 5 + 3 \cdot 6 = 32$

Linear Independence

\text{Vectors } \vec{v}_1, \vec{v}_2, \ldots, \vec{v}_n \text{ are linearly independent if } \\ c_1\vec{v}_1 + c_2\vec{v}_2 + \ldots + c_n\vec{v}_n = \vec{0} \text{ implies } c_1 = c_2 = \ldots = c_n = 0

Linear independence means no vector in the set can be expressed as a linear combination of the others.

Example: Vectors $[1, 0, 0]$, $[0, 1, 0]$, and $[0, 0, 1]$ are linearly independent.

Vector Spaces

\text{Span: } \text{span}(\vec{v}_1, \vec{v}_2, \ldots, \vec{v}_n) = \{c_1\vec{v}_1 + c_2\vec{v}_2 + \ldots + c_n\vec{v}_n : c_i \in \mathbb{R}\} \\ \text{Basis: A linearly independent set that spans the vector space} \\ \text{Dimension: The number of vectors in a basis}

Vector spaces are sets closed under vector addition and scalar multiplication.

Example: The standard basis for $\mathbb{R}^3$ is $\{[1, 0, 0], [0, 1, 0], [0, 0, 1]\}$

Eigenvalues & Eigenvectors

A\vec{v} = \lambda\vec{v} \\ \det(A - \lambda I) = 0

Eigenvalues (λ) and eigenvectors (v) are special scalars and vectors associated with a square matrix.

Example: For $A = \begin{bmatrix} 3 & 1 \\ 1 & 3 \end{bmatrix}$, eigenvalues are $\lambda = 2, 4$ with eigenvectors $\vec{v}_1 = [1, -1]$ and $\vec{v}_2 = [1, 1]$

Row Reduction

\text{Elementary Row Operations:} \\ 1. \text{Swap two rows} \\ 2. \text{Multiply a row by a non-zero scalar} \\ 3. \text{Add a multiple of one row to another}

Row reduction is used to solve systems of equations and find matrix properties.

Example: Row reducing $\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}$ to $\begin{bmatrix} 1 & 0 & -1 \\ 0 & 1 & 2 \\ 0 & 0 & 0 \end{bmatrix}$

Understanding Linear Algebra and Its Applications

What is Linear Algebra?

Linear algebra is the branch of mathematics that deals with vector spaces, linear transformations, systems of linear equations, and their representations using matrices and vectors. It provides a framework for solving problems in many areas of mathematics and its applications.

Why Linear Algebra is Important

Linear algebra forms the foundation for many fields and has numerous applications:

Key Concepts in Linear Algebra

Applications of Linear Algebra in Real-World Problems

Computer Graphics

Machine Learning and Data Science

Engineering

Economics and Finance

Common Misconceptions About Linear Algebra

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